Chapter 11
Waves
Chapter 11 Topics
• Energy Transport by Waves
• Longitudinal and Transverse Waves
• Transverse Waves on Strings
• Periodic Waves
• Superposition
• Reflection and Refraction of Waves
• Interference and Diffraction
• Standing Waves on a String
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Waves and Energy Transport
• A wave is a disturbance that travels, in a
medium, outward from its source.
• Waves carry energy.
• The energy is transported outward from the
source to the destination.
• The matter in the medium expieriences no
net transport.
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Waves and Energy Transport
When a stone is dropped into a pond, the water is disturbed
from its equilibrium positions as the wave passes; it returns
to its equilibrium position after the wave has passed.
The water moves up and
down as the disturbance
moves outward but is
not transported by the
wave.
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A Simple Water Wave
Trough
Peak
The wave travels to the right. The individual water molecules
move up and down locally but do not travel with the wave. A
surfer CAN travel with the wave, at least for a while.
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3-D Spherical Wave
Energy spreads out, from
a point source, uniformly
over a 3 dimensional
area
Intensity = Power/Area
Units: Watts/m2
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Waves and Energy Transport
Intensity is a measure of the amount of energy/sec that
passes through a square meter of area perpendicular to the
wave’s direction of travel.
Power
P
I

2
4r
4r 2
Intensity has units
of watts/m2 .
This is an inverse square law. The intensity drops as the
inverse square of the distance from the source.
(Light sources appear dimmer the farther away from them you
are.)
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Waves and Energy Transport
Example: At the location of the Earth’s upper atmosphere, the
intensity of the Sun’s light is 1400 W/m2. What is the
intensity of the Sun’s light at the orbit of the planet Mercury?
Psun
Ie 
4 res2
Psun
Im 
4 rms2
Divide one equation by the other (take a ratio):
Psun
2
2
 res   1.50 1011 m 
I m 4 rms2
  6.57

    
10
Psun
Ie
 rms   5.85 10 m 
4 res2
I m  6.57 I e  9200 W/m 2
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Longitudinal Wave
Amplitude change parallel to propagation direction
Transverse Wave
Amplitude change perpendicular to propagation direction
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The Excitation of a Transverse Wave
Boundary Condition:
The end of the string is stationary
The speed with which the string is moved vertically is
independent of the speed with the wave travels horizontally
down the string.
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Transverse Waves on a String
Attach a mass to a string to provide tension. The string
is then shaken at one end with a frequency f.
L
Attach a
wave
driver here
M
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Transverse Waves on a String
A wave traveling on this string will have a speed of
v
F

where F is the force applied to the string (tension) and
 is the mass/unit length of the string (linear mass
density).
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Transverse Waves on a String
Example (text problem 11.8): When the tension in a cord is
75.0 N, the wave speed is 140 m/s. What is the linear mass
density of the cord?
The speed of a wave on a string is
v
F

Solving for the linear mass density:
F
75.0 N
3
 2 

3
.
8

10
kg/m
2
v
140 m/s 
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Speed of Sound in Various Materials
High mass
v
F


Force
Inertia
String
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v
Low Mass
B

Liquid
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v
Y

Solid
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Periodic Waves
A periodic wave repeats the same pattern over and over.
• For periodic waves: v = f
• v is the wave’s speed
• f is the wave’s frequency
•  is the wave’s wavelength
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A Simple Harmonic Oscillator
Its motion, depicted as a function of time, is a wave.
Apeak-peak = Ap-p = 2Apeak = 2Ap; Ap = Ap-p /2
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Periodic Waves
The period T is measured by the amount of time it takes
for a point on the wave to go through one complete cycle
of oscillations. The frequency is then f = 1/T.
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Periodic Waves
One way to determine the wavelength is by measuring
the distance between two consecutive crests.
The maximum
displacement
from equilibrium
is amplitude (A)
of a wave.
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Periodic Waves
Example (text problem 11.13): What is the wavelength
of a wave whose speed and period are 75.0 m/s and 5.00
ms, respectively?
v  f 

T
Solving for the wavelength:
  vT  75.0 m/s 5.00 103 s   0.375 m
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Wave Properties
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The Principle of Superposition
For small amplitudes, waves will pass through each
other and emerge unchanged.
Superposition Principle: When two or more waves
overlap, the net disturbance at any point is the sum of the
individual disturbances due to each wave.
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Two traveling
wave pulses: left
pulse travels right;
right pulse travels
left.
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Reflection and Refraction
At an abrupt boundary between two media, a reflection
will occur. A portion of the incident wave will be
reflected backward from the boundary.
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The Reflected Wave & Phase Change
When you have a wave that
travels from a “low
density” medium to a “high
density” medium, the
reflected wave pulse will
be inverted. (180o phase
shift.)
The frequency of the
reflected wave remains the
same.
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Light Ray Example
When a wave is incident
on the boundary between
two different media, a
portion of the wave is
reflected, and a portion
will be transmitted into the
second medium. Reflected
ray is 180o out of phase.
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The Frequency is Constant
The frequency of the transmitted wave remains the
same. However, both the wave’s speed and wavelength
are changed such that:
f 
v1
1

v2
2
The transmitted wave will also suffer a change
in propagation direction (refraction).
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Example (text problem 11.36)
Light of wavelength 0.500 m in air enters the water in a
swimming pool. The speed of light in water is 0.750
times the speed in air. What is the wavelength of the
light in water?
Since the frequency is unchanged in both media:
f 
water
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vair
air

vwater
water
 vwater 
air
 
 vair 
 0.750vair 
0.500 m  0.375m
 
 vair 
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Interference
Two waves are considered coherent if they have
the same frequency and maintain a fixed phase
relationship.
Two waves are considered incoherent if the
phase relationship between them varies
randomly.
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When waves are in phase, their superposition
gives constructive interference.
When waves are one-half a cycle out of
phase, their superposition gives destructive
interference.
This is referred to as:
“exactly out of phase” or “180o out of phase.”
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Constructive
Interference
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Destructive
Interference
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Interference
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Constructive Interference. Means that the
waves ADD together and their amplitudes are
in the same direction
Destructive Interference. Means that the
waves ADD together and their amplitudes are
in the opposite directions.
Interference = Bad choice of words
• The two waves do not interfere with each other.
• They do not interact with each other.
• No energy or momentum is exchanged.
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Phase Difference
When two waves travel
different distances to reach
the same point, the phase
difference is determined
by:
d1  d 2

phase difference

2
Note: This is a ratio comparison. λ is not equal to 2π
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Diffraction
Diffraction is the spreading of a wave around
an obstacle in its path and it is common to all
types of waves.
The size of the obstacle must be similar to the
wavelength of the wave for the diffraction to
be observed.
Larger by 10x is too big and
smaller by (1/10)x is too small.
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Standing Waves
Pluck a stretched string such
that y(x,t) = A sin(t + kx)
(This is a wave that moves to the left.)
When the wave strikes the wall, there will be a
reflected wave that travels back along the string.
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The reflected wave will be 180° out of phase with the
wave incident on the wall.
Its form is y(x,t) = A sin (t  kx).
(This is a wave that moves to the right.)
Apply the superposition principle to the two waves on
the string:
y ( x, t )  y1 ( x, t )  y2 ( x, t )
 Asin t  kx  sin t  kx
 2 A cos t sin kx
The time variation and the
spatial variation have been
separated.
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The previous expression is the mathematical form of
a standing wave.
A
A
A
N
N
N
N
A node (N) is a point of zero oscillation. An antinode
(A) is a point of maximum displacement. All points
between nodes oscillate up and down.
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The nodes occur where y(x,t) = 0.
yx, t   2 A cos t sin kx  0
The nodes are found from the locations where sin kx = 0,
which happens when kx = 0, , 2,…. That is when kx
= n where n = 0,1,2,…
The antinodes occur when sin kx =  1; that is where
kx 
 3
,
,
2 2

2n  1
kx 
and n  0 ,1, 2 ,
2
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If the string has a length L, and both ends are fixed,
then y(x = 0, t) = 0 and y(x = L, t) = 0. (Boundary conditions)
y  x  0, t   sin k 0   0
y  x  L, t   sin kL  0
kL  n
2
L  n

The wavelength of
a standing wave:
2L

n
where n = 1, 2, 3,…
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2L
n 
n
These are the permitted wavelengths of
standing waves on a string; no others are
allowed.
The speed of the wave is:
The allowed frequencies are
then:
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v  n f n
fn 
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v
n

nv
2L
n =1, 2, 3,…
41
The n = 1 frequency is called the fundamental frequency.
nv
 v 
fn 

 n   nf1
n 2 L  2 L 
v
All allowed frequencies (called harmonics) are
integer multiples of f1.
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Example (text problem 11.51): A Guitar’s E-string has a
length 65 cm and is stretched to a tension of 82 N. It
vibrates with a fundamental frequency of 329.63 Hz.
Determine the mass per unit length of the string.
For a wave on a string: v 
F

Solving for the linear mass density:

F
F
F


v 2 1 f1 2 f12 2 L 2

82 N 

329.63 Hz 2 2 * 0.65 m 2
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 4.5 10  4 kg/m
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Two Methods of Classification
Just to make life interesting
1st Harmonic
Fundamental
1st Overtone
2nd Harmonic
2nd Overtone
3rd Harmonic
The labels on the right are more general. If the frequencies are integer
multiples of one another then they are referred to as harmonics
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Summary
• Energy Transport by Waves
• Longitudinal and Transverse Waves
• Transverse Waves on Strings
• Periodic Waves
• Superposition
• Reflection and Refraction of Waves
• Interference and Diffraction
• Standing Waves on a String
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Extra
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Wave Properties
The sound in an acoustic instrument comes from the
vibrating strings moving the air and coupling into the
resonant cavity of the instrument whose walls vibrate and
in turn cause vibrations in the surrounding air pressure that
we interpret as sound. It acts as a sound amplifier
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